Using Newtons Method to Solve Equations: A Step-by-Step Guide | Homework Help

In summary: If it equals 0, then your solution is correct. Otherwise, there may be a mistake in your calculations.
  • #1
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Homework Statement



Solve each equation using Newtons method. The problem I am working on right now is:

[tex] x^{5}+x^{3}+x=1 [/tex]

which is the same as:

[tex] x^{5}+x^{3}+x-1=0 [/tex]


Homework Equations



[tex] x_{2} = x_{1}-\frac {f(x)}{f'(x)} [/tex]

with x1 being the first guess to what an f(x)=0 would be.

The Attempt at a Solution



So I put the equation into my calculator to graph it and It is very clear from the graph that f(x)=0 on the x>0 side so I choose x1=1 and continue like so:

[tex] x_{2} = 1 - \frac {f(x)}{f'(x)} = 1- \frac {2}{9} = \frac {7}{9} [/tex]

and after 5 iterations I get:

[tex] x_{5} = .6368843716 [/tex]

which looks right to me but when I check the answer in the back of the book it says that it should be -.63688, so just the opposite of what I got. The graph of the equation clearly shows that f(x)=0 on x>0 so did I do something wrong or does the book have another typo?

thanks
 
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  • #2
Another typo.
 
  • #3
I've just spent the last 5 months play with Newton's method... The general method is
[tex]
x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}
[/tex]
Where in your case:
[tex]
f(x)=x^{5}+x^{3}+x-1
[/tex]
The derivative of this is:
[tex]
f'(x)=5x^{4}+3x^{2}+1
[/tex]
And in your case Newton's method is
[tex]
x_{n+1}=x_{n}-\frac{x_{n}^{5}+x_{n}^{3}+x_{n}-1}{5x_{n}^{4}+3x_{n}^{2}+1}
[/tex]
So all you have to do is plug numbers in. Newtons method converges very quickly in my experience, 2 or 3 iterations will usually do the trick.
 
  • #4
As a check, plug your solution back into the equation and check.
 

1. What is Newton's Method and how does it work?

Newton's Method is an iterative process used to find the roots of a given equation. It involves repeatedly updating an initial guess for the root based on the slope of the equation at that point, until a sufficiently accurate root is found. This method is based on the idea that the tangent line to a function at a given point will intersect the x-axis at a point closer to the actual root.

2. When is Newton's Method most useful?

This method is particularly useful when trying to find the roots of a complex or non-linear equation, as it can provide a more accurate solution than other methods such as trial and error or graphing. It is also useful when the equation does not have an algebraic solution, as it can provide a numerical approximation of the root.

3. What are the steps involved in using Newton's Method?

The steps for using Newton's Method are as follows:
1. Choose an initial guess for the root.
2. Calculate the slope of the function at that point using the derivative.
3. Use the slope and the initial guess to find the equation of the tangent line.
4. Find the x-intercept of the tangent line, which will be a closer approximation of the root.
5. Repeat steps 2-4 until the desired level of accuracy is reached.

4. What are the limitations of Newton's Method?

While Newton's Method is a powerful tool for finding roots, it does have some limitations. One limitation is that it may not always converge to the correct root, especially if the initial guess is not close enough to the actual root. Another limitation is that it can fail if the function has multiple roots or if the function is not continuous.

5. Can Newton's Method be used for solving equations with multiple variables?

Yes, Newton's Method can be extended to solve equations with multiple variables. This is known as the Newton-Raphson method and involves finding the partial derivatives of the function with respect to each variable and then using these derivatives to iteratively update the initial guess for each variable until a solution is found.

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